$ F = \left[\begin{array}{rrr}3 & 0 & 5 \\ 1 & 5 & -1\end{array}\right]$ $ E = \left[\begin{array}{rr}5 & 2 \\ -2 & -1 \\ -1 & -2\end{array}\right]$ What is $ F E$ ?
Explanation: Because $ F$ has dimensions $(2\times3)$ and $ E$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ F E = \left[\begin{array}{rrr}{3} & {0} & {5} \\ {1} & {5} & {-1}\end{array}\right] \left[\begin{array}{rr}{5} & \color{#DF0030}{2} \\ {-2} & \color{#DF0030}{-1} \\ {-1} & \color{#DF0030}{-2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rr}{3}\cdot{5}+{0}\cdot{-2}+{5}\cdot{-1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{5}+{0}\cdot{-2}+{5}\cdot{-1} & ? \\ {1}\cdot{5}+{5}\cdot{-2}+{-1}\cdot{-1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{5}+{0}\cdot{-2}+{5}\cdot{-1} & {3}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{-1}+{5}\cdot\color{#DF0030}{-2} \\ {1}\cdot{5}+{5}\cdot{-2}+{-1}\cdot{-1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{3}\cdot{5}+{0}\cdot{-2}+{5}\cdot{-1} & {3}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{-1}+{5}\cdot\color{#DF0030}{-2} \\ {1}\cdot{5}+{5}\cdot{-2}+{-1}\cdot{-1} & {1}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{-1}+{-1}\cdot\color{#DF0030}{-2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}10 & -4 \\ -4 & -1\end{array}\right] $